Congruence for rational points over finite fields and coniveau over local fields

If the ell-adic cohomology of a projective smooth variety, defined over a local field K with finite residue field k, is supported in codimension ≥ 1 then every model over the ring of integers of K has a k-rational point. For K a p-adic field, this is [8, Theorem 1.1]. If the model X is regular, one has a congruence |X (k)| ≡ 1 modulo |k| for the number of k-rational points ([7, Theorem 1.1]). The congruence is violated if one drops the regularity assumption.